Advertisements
Advertisements
प्रश्न
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
Advertisements
उत्तर
Let I = `int (x^2 + x -1)/(x^2 + x - 6) "d"x`
= `int (x^2 + x - 6 + 5)/(x^2 + x - 6) "d"x`
= `int [1 + (5/(x^2 + x - 6))] "d"x`
Let `5/(x^2 + x - 6) = 5/((x + 3)(x - 2))`
= `"A"/(x + 3) + "B"/(x - 2)`
∴ 5 = A(x − 2) + B(x + 3) ........(i)
Putting x = 2 in (i), we get
5 = B(5)
∴ B = 1
Putting x = −3 in (i), we get
5 = A(− 5)
∴ A = −1
∴ `5/((x + 3)(x - 2)) = (-1)/(x + 3) + 1/(x - 2)`
∴ I = `int[1 + (-1)/(x + 3) + 1/(x - 2)] "d"x`
= `int "d"x - int 1/(x + 3) "d"x + int 1/(x - 2) "d"x`
= x − log|x + 3| + log|x − 2| + c
∴ I = `x + log |(x - 2)/(x + 3)| + "c"`
संबंधित प्रश्न
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
`int (dx)/(x(x^2 + 1))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^5 + 1))` dx
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int x^2sqrt("a"^2 - x^6) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int 1/(2 + cosx - sinx) "d"x`
`int sec^3x "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
Evaluate `int x^2"e"^(4x) "d"x`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
What is done after long division, before writing the appropriate partial-fraction decomposition?
How are the constants \[\mathrm{A},\mathrm{B},\mathrm{C},\ldots\] determined in a partial-fraction decomposition?
Why is \[\frac{x^{2}+1}{x^{2}-5x+6}\] not a proper rational function?
Which decomposition is correct for \[\frac{x^{2}+1}{x^{2}-5x+6}\]?
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
